En muchas situaciones el significado de "entender" y "comprender" es el mismo, pero como se ha dicho antes "comprender" tiene más significados que entender. En el caso de que aparezcan los dos verbos en la misma frase ("te entiendo pero no te comprendo") "comprender" tiene un sentido más amplio. Por ejemplo, alguien me explica una situación o problema con la pareja que le ha llevado a tomar una determinada decisión. Si yo le digo a esa persona "te entiendo pero no te comprendo" quiero decir que me he enterado de la situación pero no acabo de ver claro el por qué ha tomado esa decisión. Personalmente para alguien que está aprendiendo el idioma no me molestaría en buscar diferencias. Sólo en frases hechas es importante saber que no se pueden intercambiar. Dejo abajo algunos ejemplos: Enterarse de algo ("no te enteras de nada"). Esa chica entiende (significa que esa chica es lesbiana). Me entiendo bien con el sistema operativo (significa que lo manejo bien) Entender de algo. "Entiendo de mecánica" (significa que se algo de mecánica)

Both comprender and entender are usually translated as "to understand," and in most of the time you can use them interchangeably. look at this example: "Te entiendo, pero no te comprendo," If we translated it would be "I understand you, but I don't understand you." Perhaps a better way of understanding this sentence would be something like "I understand the words you're saying but I don't understand what you mean." Can you see the difference?

Since you're a native English speaker here is the short response. Entender=understand Comprender=comprehend (Well, you can think of it this way. Comprender is used less often just like "comprehend" in English)

Despite both have the same meaning, in my opinion Entender is a more usual verb than comprender, a little bit. And there is one different meaning that I don't know if the dictionary has. We can say that one person "entiende", when this person likes people of his of her same sex. But not in a rude way, if someone doesn't want to ask directly the tastes of other person, he or she can say, "tú entiendes?"

In one of his meanings, "entender y "comprender" are the same (to understand). But each one has several meanings. Look at the RAE dictionary (http://lema.rae.es/drae/). "Comprender" has 18 differents meanings and "entender" has 4. But the main, the most usual, is the same = to understand.